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The Legendre transform, the Laplace transform and valuations

Jin Li

Abstract

We first prove that the Legendre transform is the only continuous and $\mathrm{SL}(n)$ contravariant valuation that behaves as a conjugation of two important translations on super-coercive, lower semi-continuous, and convex functions. Then we turn to a similar setting on log-concave functions and find characterizations of not merely the duality transform but also the Laplace transform on log-concave functions. With the notion of dual valuation, we also obtain characterizations of the identity transform on finite convex functions and positive log-concave functions.

The Legendre transform, the Laplace transform and valuations

Abstract

We first prove that the Legendre transform is the only continuous and contravariant valuation that behaves as a conjugation of two important translations on super-coercive, lower semi-continuous, and convex functions. Then we turn to a similar setting on log-concave functions and find characterizations of not merely the duality transform but also the Laplace transform on log-concave functions. With the notion of dual valuation, we also obtain characterizations of the identity transform on finite convex functions and positive log-concave functions.
Paper Structure (10 sections, 28 theorems, 188 equations)

This paper contains 10 sections, 28 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. The Cauchy functional equation and related results
  4. Valuations on convex bodies
  5. Translation covariant valuations
  6. Log-translation covariant valuations
  7. Main results and generalizations
  8. Generalizations of Theorem \ref{['thm:Leg']}
  9. Generalizations of Theorem \ref{['mthm:log1']} and \ref{['mthm:log']}
  10. Dual valuations and characterizations of the identity transform

Key Result

Theorem 1

Let $n \ge 2$. A map $Z:\mathscr{K}_{(o)}^n \to (\mathscr{K}^n,+)$ is a continuous valuation satisfying for every $K \in \mathscr{K}_{(o)}^n$ and $\phi \in \mathrm{GL}(n)$, if and only if there are $c_1,c_2 \ge 0$ such that for every $K\in \mathscr{K}_{(o)}^n$. Here "$+$" can be either Minkowski addition or radial addition.

Theorems & Definitions (41)

  • Theorem : Ludwig Lud06Lud10b
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 31 more